Solid Geometry—Planes

33*

lines drawn one in each plane perpendicular to the line of intersection of the
two planes.

In Fig. 167 OPQR is a rectangle ; OP and Qy are perpendicular to the
line of intersection Or of the two
planes OPQR and Qxy; the angle be-
tween the two planes is jyOP or POy.

Draw OT perpendicular to the
plane OPQR.

Then, if we suppose the plane
OPQR to lie originally in the plane
Or/, so that OT coincides with Oz,
and to rotate about Or to its present
position, OT evidently turns through
the same angle as OP, and therefore
the angle zQT == yQP - angle be-
tween the planes OPQR and Qxy.

Thus the angle between any two
planes is equal to the angle between
two perpendiculars to the respective
planes. FIG. 167.

193. Length, and Direction of the Perpendicular from the Origin
to a given Plane.—Let OP be the perpendicular from the origin to the
plane ABC,/ its length, and (/, m^ ri) its direction cosines.

FIG. x6S,
In the figure BPO is a right angle, and

m = cos yOP =^,» = cos^OP =^, / = cos rOP =^

o c a

••• ++=p+n*+n°"=i (p-322)

/./=

1

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